Essentials of Geometry

Key Concepts

• Identify complements and supplements
• Find measures of a complement and a supplement
• Find angle measures
• Identify angle pairs
• Find angle measures in a linear pair

Describe Angle Pair Relationships 

Introduction

In this chapter, we will learn to identify complementary angles and supplementary angles, find the measures of a complement and a supplement. Find the angle measures, identify angle pairs, find angle measures in a linear pair, and learn about angle pair relationships. 

Complementary angles

Two angles are complementary angles if the sum of their measures is 90^o. 

Each angle is the complement of the other. 

Supplementary angles

Two angles are supplementary angles if the sum of their measures is 180o.  

Each angle is the supplement of the other. 

Identify complements and supplements 

Example 1: 

In the figure, name a pair of complementary angles, a pair of supplementary angles, and a pair of adjacent angles. 

Solution: 

• Because 32^°+ 58^° = 90^°, ∠BAC and ∠RST are complementary angles. 
• Because 122° + 58° = 180°, ∠CAD and ∠RST are supplementary angles. 
• Because∠BAC and ∠CAD share a common vertex and side, they are adjacent. 

Find measures of a complement and a supplement 

Example 1: 

Given that ∠1 is a complement of ∠2 and m∠1 = 68°, find m∠2.

Solution: 

You can draw a diagram with complementary adjacent angles to illustrate the relationship. 

Example 2: 

Given that ∠3 is a supplement of ∠4 and m∠4 = 56°, find m∠3. 

Solution: 

You can draw a diagram with supplementary adjacent angles to illustrate the relationship. 

Find angle measures 

Example: 

Sports: When viewed from the side, the frame of a ball-return net forms a pair of supplementary angles with the ground. Find m

∠BCE and m∠ECD. 

Solution: 

Step 1: Use the fact that the sum of the measures of supplementary angles is 180°. 

Step 2: Evaluate: the original expressions when x = 34. 

m∠BCE = (4x + 8)°  = (34 + 8)°  = 144° 

m∠ECD = (x + 2)°  = (34 + 2)°  = 36° 

The angle measures are 144° and 36°. 

Angle pairs 

1. Two adjacent angles are a linear pair if their non-common sides are opposite rays.  The angles in a linear pair are always supplementary.   

∠1 and ∠2 are a linear pair. 

Let us see another example, 

2. Two angles are vertical angles if their sides form two pairs of opposite rays. 

∠3 and ∠6 are vertical angles. 

∠4 and ∠5 are vertical angles. 

Let us see another example, 

∠1=∠3

∠2=∠4

Identify angle pairs 

Example 1: 

Identify all of the linear pairs and all of the vertical angles in the figure below.  

Solution: 

• To find vertical angles, look for angles formed by intersecting lines.  
• ∠1∠1 and ∠5∠5 are vertical angles. 
• To find linear pairs, look for adjacent angles whose non-common sides are opposite rays. 
• ∠1∠1 and ∠4∠4 are a linear pair. ∠4∠4 and ∠5∠5 are also a linear pair. 

Find angle measures in a linear pair 

Example: 

Two angles form a linear pair. The measure of one angle is 5 times the measure of the other. Find the measure of each angle. 

Solution: 

Let x° be the measure of one angle. The measure of the other angle is 5x°. Then use the fact that the angles of a linear pair are supplementary to write an equation. 

Exercise

1. In the figure, name a pair of complementary angles, a pair of supplementary angles, and a pair of adjacent angles.

2. In the figure, name a pair of complementary angles, a pair of supplementary angles, and a pair of adjacent angles.

3. Given that Ðl is a complement of Ð2 and mÐl = 73°, find mÐ2.
4. Given that Ðl is a complement of Ð2 and mÐ2 = 57°, find mÐ1.
5. Given that Ð3 is a supplement of Ð4 and mÐ4 = 37°, find mÐ3.
6. Given that Ð3 is a supplement of Ð4 and mÐ4 = 41°, find mÐ3.
7. Find the values of x and y.

8. The basketball pole forms a pair of supplementary angles with the ground. Find m  and m .

9. Identify all of the linear pairs and all of the vertical angles in the figure below.

10. Identify all of the linear pairs and all of the vertical angles in the figure.

What have we learned

• To identify complementary angles and supplementary angles
• To find the measures of a complement and a supplement
• To find the angle measures
• To identify angle pairs
  • To find angle measures in a linear pair

Concept Map

Frequently asked questions 

1. What are the different types of angle pairs? 

Ans: The different types of angle pairs are Complimentary angle pairs, Supplementary angle pairs, linear angle pairs, vertical angle pairs and adjacent angle pairs. 

2. How do you identify an angle pair? 

Ans: You can identify angle pairs by the angle measurements or the angle placement. 

3. How are supplementary angles different from complimentary angles? 

Ans: Two angles are supplementary angles if the sum of their angle measures is 180°. Meanwhile its complementary angles if the sum of their angle measures is 90°.